From personal experience, yes.
No. I never saw any damage to the tank from falling trees.
If you are doing it right, they mostly don’t fall on you anyway. Hit them slow enough with he center of the front slope and you more push them over/away. It happens though. Especially if things are thick because they can hit trees behind them and bounce back towards you. The tree that landed on my head pulled that deception. Fortunately it didn’t hit me someplace vital.
There was a variety in a few of our training areas that tended to break off higher. The lower trunk on those would push away but the tree tops were like spears of doom hurled straight down onto the top armor by tree gods angry at tanks challenging their supremacy. That didn’t affect the tank either. It WAS scarier.
I did, once, mildly bend a couple of the fins on the grill over my exhaust (M1 slick). I did it by ramming into two closely set trees at high speed going backwards. ("Driver hold left!!! SMASH "Driver stop…your other left.) Admittedly one was dead but still standing. That one just exploded into sawdust and splinters. The living tree left chunks we had to pick out and some inconsequential bending to a couple fins. Going forward, taking a tree on the track tended to bend up the fender. It happened more when you were trying to weave in and around the trees to save them. That didn’t affect function. Well the left fender was a nice sleeping spot when I was still on the M60A3; not making a nice bed lumpy is kind of functional.